Analysis Of Inconsistency And Applicability Of The QAT

The concept of adiabatic approximation and adiabatic invariant came from research on obit deformation in classical mechanics. it is vitally important to the development of quantum theory. Quantum adiabatic theorem (QAT) plays an important role in the application of Schrodinger equation in quantum mechanics. Its application even exceeds the scope of quantum mechanics. However, in recent years, the consistency of QAT has been doubted continuously starting from Marslin and Sanders'article. Confusion about the validity of QAT in physics circles has been caused. Besides, physicists also have some obscure cognition about the applicability of QAT. This article makes some clarification to some wrong cognition to QAT, and meanwhile it also analyzes two typical inconsistencies of QAT to eliminate general questioning of QAT.In the article, it first briefly reviews the history of QAT and its wide applications. Then in the first chapter, this article discusses the meaning of QAT generally. We first clarify that what QAT means is that when system Hamiltonian H (t) changes slowly with time, the transition probability between different energy levels can be neglected. Then it introduces the condition of adiabatic approximation. Finally, the article introduces Berry phase in adiabatic evolution.In the second chapter, we will go over the establishment of QAT.QAT before being improved by Berry:Suppose the Hamiltonian H (t/T) depends on t/T, and its eigenvalue problem is: { u n} constitute a complete orthonormal system.【Assumption】 If there exists such a constant matrix M, for all 0≤τ≤1, and n, m. The probability of transition from u j(0) to un (t T) (for any 0≤t≤T) is:This is QAT.QAT improved by Berry:Assume that Hamiltonian depends on R(t), a set of slowly changing parameters. Work outSchrodinger equation containing time:The eigenequation of the instantaneous Hamiltonian is :Assume:For system which lies in m, R(0) at the beginning, the wave function at t is: whereThe third chapter introduces several imprecise points of view on QAT applicability, and makes some discussion. After introducing several points of view on its applicability, we re-establish QAT according to explanation need. By this explanation of'standard form', solution to adiabatic approximation has the following relationship with Schrodinger equation:Solution to adiabatic approximation that does not contain Berry phase neither meets integration form of Schrodinger equation nor meets differential form of Schrodinger equation; solution to adiabatic approximation improved by Berry approximately meets integration form of Schrodinger equation but does not meet differential form of Schrodinger equation. Next, this chapter corrects a kind of wrong understanding to QAT approximate condition, assume that Hamilton depends a series of parameters Rj , j= 1,2,...,N,simply thinking that condition of adiabatic approximation equals to is wrong, it will cause contradictory results.The fourth chapter introduces two articles about QAT inconsistency.First we will read the article by M&S [18]: it first defines a time inversion solutionWe will know initial conditionψ(t 0 )=E0(t0). Prove that the above time inversion solution meets Schrodinger equation with Hamiltonian H (t )= ?U+(t,t0 )H(t)U(t,t0); then define ( ):0 0()0(0)teEtt Etdt, prove that ? (t) approximately meets Schrodinger equation with Hamiltonian H (t )= ?U+(t,t0 )H(t)U(t,t0) and initial condition by usingQAT. Then we could conclude:Thus we could get QAT inconsistency.We will read articles [21] by Sarandy, etc:The authors prove that the adiabatic approximate solution satisfies Schrodinger equation, andUsing these, they re-obtain the M-S inconsistency. But they claim QAT is consistent. The fifth chapter makes an analysis to these two kinds of QAT inconsistencies. We could discover by analyzing that the article by M&S makes such a basic mathematics mistake, that is it thinks that functions of two time t that meets the same initial condition and approximately meets the same first order differential equation will equal to each other at any time t. we have proved that it is wrong for large t. Mistake of QAT inconsistency by Sarandy is caused by put the solution to adiabatic approximation into differential equation. Therefore, the dispute about QAT inconsistency is only some mistakes on mathematics which has nothing to do with physics.Finally, we discuss meaning of an approximate solution.ψAP ( t)is an approximate solution to integration equation or initial value problem of a differential equation only means that norm of difference between approximate solutionψAP ( t)and accurate solutionψEX ( t)is small(for accurate degree in need),ψAP ( t ) ?ψEX( t)≈0, but it does not mean that mole of derivation difference also meetsψ& EX (t )?ψ&AT(t)<<1.Thus, approximate solution to Schrodinger integration equation does not necessarily satisfy the equivalent differential Schrodinger equation. Regardingψ& AT(t)asψ& EX(t)may cause contradiction. But all physics information is contained in wave functions and has no relation with its derivation. So QAT is correct physically.